Surface operators are two-dimensional defects, which are natural generalisations of the 't Hooft operators. In [1]}, Gukov and Witten initiated a study of surface operators in \(\mathcal {N}=4\)  super Yang-Mills theories. The \(S\) -duality [2]} assert that
\(S : \; (G, \tau ) \rightarrow ( G^{L}, - 1 / n_{\mathfrak {g}} \tau ),\) 
